Note on authorship: This result was
discovered by Warren D. Smith & published on
these web pages in August 2006. We in 2011 found
out that essentially the same discovery had been made independently by Jean-Francois
Laslier in Paris and
published in an
online economics-paper-archive,
in December 2006. Laslier's analysis also is interesting and quite different from ours.

A "Condorcet Winner" is a candidate who would beat any opponent in a simple majority vote
(2-man race).
Unfortunately there are cyclic election examples where
no Condorcet Winner exists. But when one does exist, it seems difficult
(butpossible!)
to dispute that
he ought to be the election winner. And so, much fear and gnashing of teeth
arose from the realization that the
Approval Voting
system (and hence the Range Voting system) could fail to elect a Condorcet Winner.

But we shall now see that, in practice under reasonable assumptions about the strategic
behavior of approval voters,
Approval
elections will choose Condorcet winners whenever they exist, and in fact (counterintuitively!),
plausibly will do so better in practice than "official" Condorcet
voting methods!
(See also puzzle #61 for
another reason that
might be so.)

And because strategic range voters generally vote approval-style, the same would
be true of range voting elections with strategic voters.
In other words, to the extent range voters are strategic they will elect
Condorcet winners; whereas to the extent they are honest,
range voting should perform better than Condorcet.
Even if you don't quite buy all that, we think you still will agree that
in practice, one should expect no great advantage for Condorcet methods over the
simpler range voting system.

The reasonable assumption:
Approval voters act according to the
following strategy: they order the candidates from best-to-worst,
then select a "threshold" T, and they approve the candidates above T.
They choose T to cause their vote to have the most impact.

The claim:
Let N≥2.
There does not exist an N-candidate tie-free election
in which the Approval (A) and Condorcet (C) winners differ, provided
that, if they were going to differ (A≠C) that all the approval voters would
place their threshold T strategically under the assumption the winner was going
to be either A or C, i.e. would place T somewhere between them.

Proof:
Assume for a contradiction that A≠C. Then the approval voters
will strategically place their thresholds
between C and A.
But that will cause C to be approved more times than A is approved (since C, being
the Condorcet winner, is preferred over A by a majority). Hence the claim A
was the Approval winner and A≠C, leads to a contradiction. Hence either A=C
or there is no Condorcet winner C.
Q.E.D.

Remark:
Actually, essentially the same argument might seem to
show that Condorcet winners (when they exist),
will be elected by strategic voters not only under Range and Approval voting, but
in fact under a large variety of voting systems.
When we investigate that, though,
we find that the argument works better for range than for the three
most-common alternative voting systems.
That analysis
subpage
tries to gain more understanding of
exactly which
voting systems work and under exactly which assumptions about
strategic voter behavior – and exactly how "reasonable" are those assumptions? –
and exactly what are the experimental numbers?

Conclusions (some counterintuitive)

In view of this fact, it seems that much of the so-called "conflict"
between the Approval and Condorcet "philosophies," is illusory.

We now see that approval voting always will elect the
Condorcet Winner C (if C exists) provided merely that the voters
consider it likely that both C and whoever (A) the putative non-C approval winner
might be, might be elected – and hence vote strategically
about C versus A.

So we see that for practical purposes, Approval is a Condorcet method!

Indeed – counterintuitively – it might actually be that
Approval Voting is more likely to elect the Condorcet Winner in practice,
than Condorcet methods! (Indeed, experiments
indicate that
happens.)

Why?
Because in approval voting it is quite rare that strategically voting
dishonestly,
is wise. (And when it is wise, it is even
rarer
that people will actually realize it and do it.)
In other words, with Approval, people will tend to honestly order the candidates, and
the only strategic decision they'll make is where to locate their approval
"threshold."

In contrast, with Condorcet methods with rank-order ballots, it seems a lot more
common
to see a way to be usefully strategic-dishonest in your vote.
(See also puzzle #62.)
In other words, I suspect
it will be comparatively common that voters will dishonestly misorder their preferences
to, e.g. try to elect lesser evils.

In that case, it might well be that Condorcet rank-order ballot systems will do
something
silly (like not electing
whoever should have been the Condorcet Winner with
honest votes, such as in this example)
more commonly than Approval will in practice fail to elect the Condorcet Winner.

In that case, for practical purposes Approval will be a better Condorcet method than
actual "official" Condorcet methods!

I really think this is actually quite likely, it is not just some unlikely theoretical
speculation.
And if so, it is quite a remarkable counterintuitive conclusion.